Correspondence to: [email protected] and more accurate that presented by Baronia and Tiwari (1999); Dwivedi et al., (2002). The dispersion relation given in eq

International Journal of Scientific & Engineering Research, Volume 7, Issue 2, February-2016 1438 ISSN 2229-5518

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Small-scale kinetic Alfven waves in the presence of warm and cold magnetized plasma in plasma sheet boundary layer

– kinetic approach Jaya1 Shrivastava, Ganpat Ahirwar2 and Geeta Shrivastava2

Department of Physics, Dronacharya group of Institutions, Greater Noida (U.P.) – INDIA

Department of Physics, Vikram University, Ujjain (M.P.) - INDIA

Correspondence to: [email protected]

Abstract – Kinetic Alfven wave has been studied by kinetic approach to evaluate the dispersion relation and damping- rate of small scale Alfven wave in homogenous plasma in both cases (warm and cold electron limit) in plasma sheet boundary layer. Kinetic effects of electrons and ions are included to study kinetic Alfven wave because both are important in the transition region. It is found that the ratio β of electron thermal energy density to magnetic field energy density and the ratio of ion to electron thermal temperature (Ti/Te) affect the dispersion relation and damping-rate in both cases (warm and cold electron limit). It is predicted that higher k⊥ρi >1 are more effective for the bi-Maxwellian distribution, here k⊥ is perpendicular wave number and ρi is the ion gyro radius. The method may be suitable to deal with the PSBL, where particle acceleration is also important along with wave emissions.

Key-words: Kinetic Alfven wave, plasma sheet boundary layer, magnetospheric plasma, particle acceleration

—————————— —————————— 1. Introduction

Kinetic Alfven waves are of great importance in laboratory and space plasmas.

These waves may play an important role in energy transport, in driving field-aligned

currents, in particle acceleration and heating, and in explaining inverted-V structures in

magnetosphere-ionosphere coupling and in solar flares and the solar wind (Hasegawa,

1976; Goertz and Boswell, 1979; Goertz, 1984). Additional support for their role in

auroral phenomena comes from global distribution maps at both low (FAST satellite) and

high (Polar satellite) altitude show that Alfven waves occur on auroral field lines along

the entire auroral oval (Chaston et al., 2003a; Keiling et al., 2001, 2003, 2005).

Hasegawa (1976) first suggested that small-scale kinetic Alfven waves possessed

parallel electric fields and could be an efficient mechanism for accelerating particles on

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plasma sheet field lines. The kinetic Alfven waves Hasegawa discussed had spatial scales

perpendicular to B such that k⊥ρS ~ 1, where ρS is the ion acoustic gyroradius. This is the

“kinetic” limit of kinetic Alfven waves. Subsequent theoretical investigations (Goertz and

Boswell, 1979) focused on kinetic Alfven waves in the electron inertial limit (k⊥ c/ωpe

~1) as a mechanism for establishing parallel electric fields capable of accelerating auroral

electrons. Lysak and Lotko (1996) included the effects due to small perpendicular scale

sizes of the order of the electron inertial scales (c/ωpe) or kinetic effects associated with

electron pressure and ion acoustic gyroradius. These effects are called “kinetic Alfven

waves”.

Recent polar/Hydra (Scudder et al., 1995), electric field instrument (Harvey et al.,

1995) spacecraft observations (Wygant et al., 2000; 2002) at altitudes of 4–7 RE in the

PSBL and deeper within the plasma sheet show that large amounts of energy are

transferred earthward by the Poynting flux carried by Alfveń waves. This enhanced

Alfveńic Poynting flux dominates other forms of energy flux along plasma sheet

magnetic field lines and coincides with magnetically conjugate intense aurora. The

Poynting flux due to the steady state magnetic field perturbations associated with the

field-aligned currents and the convection electric field was estimated to be ~1-2 orders

less than the Alfveńic Poynting flux. Polar/Hydra (Scudder et al., 1995), electric field

instrument (Harvey et al., 1995) spacecraft observations (Wygant et al., 2000; Keiling et

al., 2001, 2003, 2005) provide evidence that the Alfveńic Poynting flux is responsible

for transferring the power needed for the acceleration of all the energized auroral particle

populations accelerated into the ionosphere and also streaming out into the

magnetosphere, as well as the Joule heating of the ionosphere.

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In past, kinetic Alfven waves have been analyzed in view of the auroral

acceleration process at much lower altitudes of about 1.4 RE (Baronia and Tiwari, 1999;

Dwivedi et al. 2002) using S3-3 satellite data (Tiwari and Rostoker, 1984). However, it is

only recently, with the availability of the necessary measurements from the Polar /Hydra

(Scudder et al., 1995), electric field instrument (Harvey et al., 1995) spacecraft (Wygant

et al., 2000; Keiling et al., 2001, 2003, 2005), that a relatively complete inventory of the

important energy transfer mechanisms along plasma sheet magnetic field lines above the

bulk of the auroral acceleration region has become possible. The kinetic effect is true at

higher altitudes and inertial effect at lower altitudes. The observations in the auroral

acceleration region (Chaston et al., 2003 a,b; Chaston 2004; Chaston et al., 2005,

2006,2007), suggest that at altitudes of about 2-3 RE where much of the auroral

acceleration is thought to occur (Goertz and Boswell, 1979) and it has been shown that

the perpendicular wavelength is closer to the electron inertial length (c/ωpe) (Louarn et

al.,1994; Chaston 1999) than the ion acoustic gyro-radius. Under such conditions the

electrons can no longer be considered massless and so this wave will so carry a parallel

electric field. This has led to a variety of models that have considered the development of

parallel electric fields due to electron inertia (Seyler, 1988; Rönnmark and Hamrin, 2000;

Seyler and Wu, 2001

In this paper, kinetic effects of electrons and ions are included to study kinetic

Alfven wave because both are important in the transition region. The wave propagating

obliquely to the ambient magnetic field B0 is considered. The organization of the paper is

as follows. An introduction is given in Section 1. Section 2 deals with the theory and

results and conclusion are presented in Sect. 3 and 4.

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2. Theory

2.1 Basic Assumption

To determine the dispersion relation and damping rate, zeroth-order distribution

function is adopted suitable for drifting core plasma in plasma sheet boundary layer

(Horwitz et al., 1987, 2000) and is given by

(1)

Here,

Where, TΠ and T⊥ are the parallel and perpendicular temperatures with respect to

the ambient magnetic field. The “core” of magnetospheric plasma distribution functions

is of prime importance in the magnetospheric plasma behavior. Horwitz et. al. (1987,

2000) described the ionosphere as the basic core plasma region for the overall

magnetosphere-ionosphere system. They described the principal inner/middle

magnetospheric regions-the plasmasphere, ring current, and plasma sheet boundary layer

regions-and how core plasmas from the ionosphere, either with little or with substantial

energization, become major components of these magnetospheric regions.

2.2 General dispersion relation for kinetic Alfven wave

−

=

⊥

⊥

⊥⊥⊥ T

mT

mf2

exp2

)(2VV

π

−

=

∏

∏

∏∏∏ T

mVT

mVf2

exp2

)(22

1

π

0 ( ) ( )jF n f f V⊥ ⊥= ∏ ∏V

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Plasma with an external magnetic field B in the z-direction is considered and

since the KAW has its electric vector and wave vector in the same plane, its dispersion

relation can be written in matrix form as (Lysak and Lotko, 1996)

2 2 2k k k

2 2ω ω2 2 2k k k

2 2ω ω

C C

C C

ε

ε

∏ ⊥ ∏−⊥

⊥ ∏ ⊥−

x

z

EE

=0 (2)

Where k⊥ and kΠ are the components of the wave vector k across and along the

static magnetic field. k⊥ and kΠ are taken to be positive and ω is the wave frequency. All

other symbols have their usual meanings. zzε and xxε are dielectric tensor elements

among and across external magnetic field and given as:

( )

( )

22

223 k

1 k

n jjpj

xx j j jj n j

n Jv vd vv F F F

v v v vn v

µµω

εω ωω

∞∏ Π ⊥

⊥=−∞ ⊥ ⊥ Π Π∏ Π

∂ ∂ ∂ = + − − ∂ ∂ ∂− Ω −

∑ ∑ ∫

(3)

( )( )

223 n

1 k

n jpj jzz j j j

j n j

J vd vv F F Fv v v vn v

µωε

ω ωω

∞Π

Π=−∞ Π ⊥ ⊥ Π∏ Π

Ω ∂ ∂ ∂= + + − ∂ ∂ ∂− Ω −

∑ ∑ ∫

(4)

Where ω2pj=4πN0ej

2/mj is the square of plasma frequency of species j, µ j=k2⊥ρ2

j, ρi(ρe)

is the thermal ion(electron) gyroradius and Ω j= ejB0/mjc is the gyro frequency.

Substituting the value of Fj from eq.(1), zzε and xxε can be written as

( )22

2

( )1 pj n j

xx njj n j j

nZ

k vω µωε χω µ

∞

=−∞

Γ= +∑ ∑

(5)

( )2

21 ( )pjzz nj n j nj

j nj

Zk v

ω ωε χ µ χω

∞

=−∞

′= + Γ∑ ∑

(6)

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Where χnj= ω/kVj, Vj= (2Tj/mj) 1/2,λ2De =(ε0Te/ne2)1/2 is the Debye length, Z(χnj) is the

plasma dispersion function and Γn(µj)=e-µj In(µj) is the modified Bessel function. By

expending Z function and it’s derivative, zzε and xxε can be written as

02 2

( )1 1 ( )k

ezz

De

Zµε χ χλ∏

Γ= + + (7)

2

02

1 ( )1 ixx

A i

cV

µεµ

−Γ= + (8)

The dispersion relation is found by taking the determinant of the matrix in

equation (2) and is given as:

(9)

This equation can be solved numerically and the dispersion relation written in form:

(10)

If electron gyroradius is small, Γ0(µe)≈1. Eq. (6) can be simplified by taking

VA2«c2 and k2

Πλ2De «1 so that it becomes as

(11)

For warm electron limit (χne«1); expanding the Z function (Shrivastava &

Shrivastava; 2008), the dispersion relation can be written as

2 2

22 2 2 20 0

2

2 kω 11 ( )k ( ) 1 ( ) k

s

iAA e De

i

VV Zc

ρµ µ χ χ λ

µ

⊥

∏ ∏

= +−Γ Γ + ++

2 2

0 0 02 2 2 2 2 2

0 0

2( ) 1 ( ) 1 ( )ω ω ω1 1 ( ) 1k ( ) k k V ( )

e i i

s i A i T i i

ZC V

µ µ µχ χµ µ µ∏ ∏ ∏ ⊥

Γ −Γ −Γ − + − = Γ Γ

2 2

2 20 0

2 kωk 1 ( ) ( ) 1 ( )

i s

A i eV Zµ ρ

µ µ χ χ⊥

∏

= +−Γ Γ +

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(12)

For cold electron limit (χne»1), the dispersion relation can be written as

(13)

The dispersion relation of kinetic Alfven wave in warm and cold electron limit is

similar to that derived by Shrivastava & Shrivastava (2008) using particle aspect

approach and more accurate that presented by Baronia and Tiwari (1999); Dwivedi et al.,

(2002). The dispersion relation given in eq. (10) for kinetic Alfven waves includes full

kinetic effects of ions and electrons in homogeneous plasma by kinetic approach and may

be very useful for transition region where both effects are important as observed by

polar/Hydra (Scudder et al., 1995), electric field instrument (Harvey et al., 1995).

For small Larmor radius limit (k⊥ρi <1) of ions, eq.(8) can be written as:

2

2

31 14xx i

A

cV

ε µ = + −

Hence, the dispersion relation of kinetis Alfven wave is obtained as

(14)

( ( ))( )2

2 2 22 2

ω 1 1k i s

A

k iV

ρ ρ π χ⊥∏

≈ + + −

( )

+

+≈

⊥

⊥

∏

2

22

22

22

2

1

1k

ω

pe

i

A ck

kV

ω

ρ

22 2

2 2

ω 31k 4

ei

A i

Tk

V Tρ⊥

∏

≈ + +

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For large Larmor radius limit (k⊥ρi >1) of ions, eq.(8) can be written as:

2

2

11xxA i

cV

εµ

= +

Hence, the dispersion relation of kinetis Alfven wave is obtained as

(15)

Where

The dispersion relation of kinetic Alfven wave in both regimes is similar to that

derived by Sallimullah & Rosenberg (1999) if the effect of dust is neglected.

2.3 Damping-rate

Imaginary term in eq.(8) gives the landau damping of the wave.

Assuming ω ω i lγ→ + , with lγ ω , an expression for the collisionless damping rate of

the kinetic Alfven wave in the both regimes is obtained as

(12)

The expression for the damping rate of the kinetic Alfven wave is same as derived

by Sallimullah & Rosenberg (1999) if the effect of dust particles is neglected.

22 2

2 2

ω 1k

ei

A i

Tk

V Tρ⊥

∏

≈ +

speed.n Alfve' of square theis ω

cpi

2i

222 Ω=AV

2 2

28e

ei pi

m c k k Vl mπγ

ω⊥

= −

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3. Results

In the numerical evaluation of the dispersion relation, associated currents per unit

wavelength, parallel energy of resonant electrons per unit wavelength and

growth/damping rate, the following plasma parameters are used suitable for the transient

region at 4-7 RE in plasma sheet boundary layer (Wygant et al., 2000, 2002; Keiling et

al., 2000, 2001, 2002, 2003, 2005; Shrivastava & Shrivastava; 2008; Marklund et al.,

2001; Hull et al., 2010).

B0 =400nT; c/ωpe =10 km; kT⊥i = 4 keV; n0 =0.3 cm-3; ρi =20km; ρe =0.3km.

Two parameters are important in understanding the propagation of the kinetic

Alfven waves in plasma sheet boundary layer. The ratio of electron thermal energy

density to magnetic field energy density β and the ratio of ion to electron thermal

temperature (Ti/Te) control whether there is significant Landau damping of the waves

(Wygant et al., 2000, 2002). At the edge of the plasma sheet, β is ∼.001 which is

comparable to me/mi (the electron to ion mass ratio) as discussed by Wygant et al.

(2000). In the present case β is fluctuating but on the order of .001. The Polar/Hydra

measurements (Scudder et al., 1995) also indicate the ratio of ion to electron thermal

temperature is Ti/Te=2. The average plasma density 0.3 cm-3 as observed by the Polar/

electric field instrument (Harvey et al., 1995; Wygant et al., 2000, 2002), considered in

the present analysis.

Fig. 1 shows the variation of kinetic Alfven wave frequency ω with k⊥ρi for

different values of β (in the case of warm electron limit). The dispersion relation given in

eq. (6) is similar to that derived by Lysak and Song (2003), Shrivastava & Shrivastava

(2008) for the warm electron limit. It is noticed that the phase velocity of the wave

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increases with k⊥ρi and β is more effective towards the higher perpendicular wave

numbers (k⊥ρi >1). Furthermore, it is observed that the phase velocity is decreased at the

higher values of β, which may be due to the increase of electron density. Wygant et al.

(2002) presented the frequency range 0.17 to 4 Hz based on polar/Hydra (Scudder et al.,

1995), electric field instrument (Harvey et al., 1995) spacecraft observations and which is

similar as discussed in the presented paper. Wygant et al. (2000) indicated that small

scale Alfven structures may undergo Landau damping and thereby accelerate electrons

through small parallel electric fields and which is presented in fig. 2.

Fig. 3 shows the variation of kinetic Alfven wave frequency ω with k⊥ρi for

different values of Ti/Te for the warm electron limit. It is noticed that the phase velocity

increases with k⊥ρi and is higher at Ti/Te =.1. Furthermore, it is also noticed that the

phase velocity of the wave is decreased at the higher values of Ti/Te, which may be due

to the decrease of electron temperature. Fig. 4 shows the variation of damping rate γ/ω

with k⊥ρi for different values of Ti/Te. It is noticed that damping rate γ/ω is negligible

when electron temperature is higher as compared to ion temperature. It means small scale

Alfven structures may undergo Landau damping and thereby accelerate electrons through

small parallel electric fields when ion temperature is higher as compared to electron

temperature. The Polar/Hydra measurements (Scudder et al., 1995) also indicate the ratio

of ion to electron thermal temperature is Ti/Te=2.

Fig. 5 shows the variation of kinetic Alfven wave frequency ω with k⊥c/ωpe for

different values of β for the cold electron limit. The dispersion relation given in eq. (7) is

similar to that derived by Lysak and Song (2003) using kinetic approach for the cold

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electron limit. It is observed that the phase velocity increases with k⊥c/ωpe but decreases

withβ.

4. Conclusion

The present study predicts that the wave frequency and damping/growth-rate in

both cases (warm and cold electron limit) are influenced by the ratio of electron thermal

energy density to magnetic field energy density β and the ratio of ion to electron thermal

temperature (Ti/Te). It is predicted that higher k⊥ρi (k⊥ρi >1) is more effective for the bi-

Maxwellian distribution. The present model also predicts that there is no landau damping

occurred in the case of cold electron limit hence the expressions obtained for warm

electron limit are more appropriate for polar observations. Recent observations as well as

theoretical considerations presented here suggest that Alfven waves occurred in the

upward current region (primary current) and can be efficient accelerators of auroral

electrons. Landau dissipation in plasma sheet can convert significant amounts of wave

energy into the field-aligned acceleration of auroral electrons. While the calculations

presented here only give a preliminary picture of these wave-particle interactions, they

provide the basis for more sophisticated further studies of this interaction.

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Caption to figures: Fig. 1 ω versus k⊥ρi for different β at fixed ratio of ion to electron thermal temperature

Ti/Te =2.

Fig.2 Damping rate versus k⊥ρi for different β at fixed ratio of ion to electron thermal

temperature Ti/Te =2.

Fig.3 ω versus k⊥ρi for different ratio of ion to electron thermal temperature Ti/Te at

fixed β =.001.

Fig.4 Damping rate versus k⊥ρi for different ratio of ion to electron thermal temperature

Ti/Te at fixed β =.001.

Fig.5 ω versus k⊥c/ωpe for different β at fixed ratio of ion to electron thermal temperature

Ti/Te =2.

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Correspondence to: [email protected] and more accurate that presented by Baronia and Tiwari (1999); Dwivedi et al., (2002). The dispersion relation given in eq